Find the critical numbers of f (if any). Find the open...

Find the critical numbers of $f$ (if any). Find the open intervals on which the function is increasing or decreasing and locate all relative extrema. Use a graphing utility to confirm your results.
$$
f(x)=x^{2 / 3}-4
$$

Key Concepts

Critical Numbers

Critical numbers are points in the domain of a function where its derivative is zero or undefined. They are essential in identifying potential local maximum or minimum points, as changes in the sign of the derivative around these numbers indicate shifts in the function's behavior.

Derivative

The derivative of a function provides information about its rate of change. It is calculated using differentiation rules such as the power rule and is used to determine both where the function's slope is zero (finding critical numbers) and where the function is increasing or decreasing.

Increasing and Decreasing Intervals

An interval on which a function is increasing is where its derivative is positive, while it is decreasing on intervals where the derivative is negative. This analysis helps in understanding the monotonic behavior of the function and is instrumental in locating regions of local extrema.

Relative Extrema

Relative extrema, including local minima and local maxima, are points where a function reaches a peak or a trough in a specific neighborhood. These are typically identified by analyzing the sign changes of the derivative around the critical numbers.

Graphical Utility

Graphing utilities are valuable tools for confirming analytical findings. By plotting the function and its derivative, one can visually inspect the behavior of the function, verify the intervals of increase and decrease, and pinpoint the locations of relative extrema.

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